INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 38 (2005) 683–696 doi:10.1088/0305-4470/38/3/008 Separability dynamics of two-mode Gaussian states in parametric conversion and amplification A V Dodonov 1 , V V Dodonov 2 and S S Mizrahi 1 1 Departamento de F´ ısica, Universidade Federal de S˜ ao Carlos, Via Washington Luiz Km 235, S˜ ao Carlos, 13565-905, SP, Brazil 2 Instituto de F´ ısica, Universidade de Bras´ ılia, Caixa Postal 04455, 70910-900 Bras´ ılia, DF, Brazil E-mail: adodonov@df.ufscar.br, vdodonov@fis.unb.br and salomon@df.ufscar.br Received 10 August 2004, in final form 8 November 2004 Published 23 December 2004 Online at stacks.iop.org/JPhysA/38/683 Abstract We give a simplified form of Simon’s separability criterion for two-mode Gaussian states, showing that for systems whose unitary evolution is governed by arbitrary time-dependent quadratic Hamiltonians, the separability dynamics is completely described in terms of the determinant of the cross-covariance matrix. As concrete examples, we consider the evolution of the ‘inverse negativity coefficient’ (which gives a quantitative estimation of the ‘degree of entanglement’) for two initially uncoupled modes (each being in a squeezed thermal state) in the cases of parametric converter, parametric amplifier and for a cavity whose boundary oscillates in resonance with two field modes. PACS numbers: 03.65.Ud, 03.67.Mn, 42.50.Dv, 42.65.Yj 1. Introduction Various problems related to entangled quantum states were subjects of numerous studies performed over the past decade [1–4]. One of them is the condition of separability of mixed quantum states, i.e., a possibility of representing the statistical operator ˆ ρ of the total system as a sum of direct products of statistical operators acting on each part separately: ˆ ρ =  i p i ˆ ρ i 1 ⊗ ˆ ρ i 2 p i  0  i p i = 1. (1) Recently, this problem was solved for bipartite continuous variable Gaussian states [4–11]. In the most explicit form, the separability criterion was given by Simon [6]. The aim of our paper is to show that Simon’s criterion can be significantly simplified, if the system under consideration does not interact with any dissipative environment, and its dynamics is governed by an arbitrary quadratic Hamiltonian. It turns out that this criterion is closely related to the concept of universal quantum invariants introduced in [12, 13]. For this 0305-4470/05/030683+14$30.00 © 2005 IOP Publishing Ltd Printed in the UK 683